43:
102:
1423:
1179:
2169:
4639:
1908:
3016:
1418:{\displaystyle {\begin{aligned}\mathbf {1} _{A\cap B}&=\min\{\mathbf {1} _{A},\mathbf {1} _{B}\}=\mathbf {1} _{A}\cdot \mathbf {1} _{B},\\\mathbf {1} _{A\cup B}&=\max\{{\mathbf {1} _{A},\mathbf {1} _{B}}\}=\mathbf {1} _{A}+\mathbf {1} _{B}-\mathbf {1} _{A}\cdot \mathbf {1} _{B},\end{aligned}}}
2425:
4476:
1899:
640:
2877:
4459:
2308:
4317:
of the domain given by the positive half-line. In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the
Heaviside step function naturally generalises to the indicator function of some domain
2890:
2773:
2164:{\displaystyle \mathbf {1} _{\bigcup _{k}A_{k}}=1-\sum _{F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|}\mathbf {1} _{\bigcap _{F}A_{k}}=\sum _{\emptyset \neq F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|+1}\mathbf {1} _{\bigcap _{F}A_{k}}}
3739:
2581:
2505:
1761:
523:
1536:
4634:{\displaystyle -\int _{\mathbb {R} ^{n}}f(\mathbf {x} )\,\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}\;d^{n}\mathbf {x} =\oint _{S}\,f(\mathbf {\beta } )\;d^{n-1}\mathbf {\beta } .}
3822:
1691:
3359:
4375:
1184:
513:
4305:
3655:
4191:
2698:
4241:
4122:
3265:
3207:
3097:
2625:
4370:
4027:
736:
3149:
2787:
428:
264:
191:
3972:
2538:
1584:
1109:
1048:
4833:
2252:
1074:
2289:
767:
293:
2654:
4915:
3930:
3462:
3429:
3396:
1013:
353:
4977:
827:
1617:
965:
220:
4871:
4062:
3903:
3868:
1727:
323:
4942:
1754:
1476:
2201:
1174:
3842:
3011:{\displaystyle \operatorname {Cov} (\mathbf {1} _{A}(\omega ),\mathbf {1} _{B}(\omega ))=\operatorname {P} (A\cap B)-\operatorname {P} (A)\operatorname {P} (B)}
1449:
1151:
1131:
679:
891:
860:
2420:{\displaystyle \operatorname {E} (\mathbf {1} _{A})=\int _{X}\mathbf {1} _{A}(x)\,d\operatorname {P} =\int _{A}d\operatorname {P} =\operatorname {P} (A).}
2710:
1481:
830:
31:
3660:
1623:
5351:
5217:
5163:
5123:
3476:
is "true" or satisfied", plays a useful role in Kleene's definition of the logical functions OR, AND, and IMPLY, the bounded- and unbounded-
2543:
2466:
465:
5031:(Sixth reprint, with corrections ed.). Netherlands: Wolters-Noordhoff Publishing and North Holland Publishing Company. p. 227.
4688:
853:
4246:
4727:
4135:
3523:
86:
64:
3033:
in his 1934 paper "On undecidable propositions of formal mathematical systems" (the "¬" indicates logical inversion, i.e. "NOT"):
4196:
4077:
3744:
1894:{\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}})=\mathbf {1} _{X-\bigcup _{k}A_{k}}=1-\mathbf {1} _{\bigcup _{k}A_{k}}.}
635:{\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1~&{\text{ if }}~x\in A~,\\0~&{\text{ if }}~x\notin A~.\end{cases}}}
3557:
3511:
3299:
2441:
5192:(Sixth reprint, with corrections ed.). Netherlands: Wolters-Noordhoff Publishing and North Holland Publishing Company.
374:
898:. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.
5173:
4997:
3468:. What appears to the modern reader as the representing function's logical inversion, i.e. the representing function is
3274:
875:
for the function defined here almost exclusively, while mathematicians in other fields are more likely to use the term
5361:
5356:
5346:
5236:
4737:
4683:
849:
3583:
5336:
5149:
4742:
4673:
4648:
4331:
3547:
2659:
4125:
3212:
3154:
3044:
2586:
4337:
895:
852:. (This must not be confused with "dummy variables" as that term is usually used in mathematics, also called a
57:
51:
2872:{\displaystyle \operatorname {Var} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)(1-\operatorname {P} (A))}
5331:
4693:
4454:{\displaystyle \delta _{S}(\mathbf {x} )=-\mathbf {n} _{x}\cdot \nabla _{x}\mathbf {1} _{\mathbf {x} \in D}}
4072:
3977:
3535:
1428:
699:
3102:
225:
152:
109:): the "raised" portion overlays those two-dimensional points which are members of the "indicated" subset (
5341:
4747:
4712:
3935:
2510:
2430:
914:
838:
138:
68:
1543:
1079:
1018:
4802:
3515:
2259:
2237:
1053:
2265:
743:
269:
5137:
5076:
4129:
868:
556:
5155:
2630:
2212:
933:
105:
A three-dimensional plot of an indicator function, shown over a square two-dimensional domain (set
101:
5205:
5092:
5066:
5057:
Lange, Rutger-Jan (2012). "Potential theory, path integrals and the
Laplacian of the indicator".
4876:
3908:
3561:
3434:
3401:
3368:
2223:
864:
433:
2449:
989:
328:
4947:
3552:
In general, the indicator function of a set is not smooth; it is continuous if and only if its
805:
5255:
5213:
5159:
5119:
4717:
4698:
4466:
3507:
2461:
2231:
2218:
As suggested by the previous example, the indicator function is a useful notational device in
925:
887:
134:
1593:
938:
196:
30:
This article is about the 0-1 indicator function. For the 0-infinity indicator function, see
5307:
5299:
5271:
5263:
5245:
5141:
5133:
5084:
4846:
4032:
3873:
3847:
3573:
3498:
1706:
298:
4920:
1732:
1454:
5267:
5201:
4732:
4708:
4703:
4068:
3531:
3519:
2292:
834:
646:
437:
362:
2176:
5080:
1156:
5227:
5185:
5145:
5024:
4770:
3827:
3569:
3553:
3270:
3022:
Characteristic function in recursion theory, Gödel's and Kleene's representing function
2296:
1434:
1136:
1116:
17:
5250:
5231:
652:
5325:
5303:
5283:
5197:
4668:
3565:
3026:
2445:
2219:
979:
141:
that maps elements of the subset to one, and all other elements to zero. That is, if
5096:
2452:. (See paragraph below about the use of the inverse in classical recursion theory.)
2440:, the inverse of the indicator function may be defined. This is commonly called the
4652:
3502:, characteristic functions are generalized to take value in the real unit interval
2768:{\displaystyle \operatorname {E} (\mathbf {1} _{A}(\omega ))=\operatorname {P} (A)}
2437:
5088:
4678:
3477:
2204:
883:
441:
118:
5312:
3577:
2883:
971:
845:
5259:
2444:, as a generalization of the inverse of the indicator function in elementary
4836:
3488:
In classical mathematics, characteristic functions of sets only take values
295:
is a common notation for the indicator function. Other common notations are
3734:{\displaystyle V=\left\{x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\right\}}
5275:
4722:
2780:
4309:
Thus the derivative of the
Heaviside step function can be seen as the
2576:{\displaystyle \mathbf {1} _{A}\colon \Omega \rightarrow \mathbb {R} }
2500:{\displaystyle \textstyle (\Omega ,{\mathcal {F}},\operatorname {P} )}
3522:). Such generalized characteristic functions are more usually called
130:
4334:
gives rise to a 'surface delta function', which can be indicated by
1531:{\displaystyle \mathbf {1} _{A^{\complement }}=1-\mathbf {1} _{A}.}
5071:
100:
2222:. The notation is used in other places as well, for instance in
2703:
36:
4777:
3296:
For example, because the product of characteristic functions
879:
to describe the function that indicates membership in a set.
4808:
4473:. This 'surface delta function' has the following property:
3530:
sets. Fuzzy sets model the gradual change in the membership
2522:
2482:
4776:
appears because it is the initial letter of the Greek word
3817:{\textstyle P(x)=\prod \left(1-f_{\alpha }(x)^{q-1}\right)}
628:
1686:{\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}}(x))}
4711:, a function that can be viewed as an indicator for the
4067:
Although indicator functions are not smooth, they admit
5154:(Second ed.). MIT Press and McGraw-Hill. pp.
5116:
Real
Analysis: Modern Techniques and Their Applications
3354:{\displaystyle \phi _{1}*\phi _{2}*\cdots *\phi _{n}=0}
521:
3747:
2470:
841:
of the standard definition of the indicator function.
4950:
4923:
4879:
4849:
4805:
4479:
4378:
4340:
4249:
4199:
4138:
4080:
4035:
3980:
3938:
3911:
3876:
3850:
3830:
3663:
3586:
3437:
3404:
3371:
3302:
3215:
3157:
3105:
3047:
2893:
2790:
2713:
2662:
2633:
2589:
2546:
2513:
2469:
2311:
2268:
2240:
2179:
1911:
1764:
1735:
1709:
1626:
1596:
1546:
1484:
1457:
1437:
1182:
1159:
1139:
1119:
1082:
1056:
1021:
992:
941:
808:
746:
702:
655:
526:
468:
377:
331:
301:
272:
228:
199:
155:
3273:
offers up the same definition in the context of the
1703:
s. This product has the value 1 at precisely those
5148:(2001). "Section 5.2: Indicator random variables".
4843:. Consequently, both sets are sometimes denoted by
508:{\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\}}
27:
Mathematical function characterizing set membership
5004:. New York, NY: Raven Press Books. pp. 41–74.
4971:
4936:
4909:
4865:
4827:
4633:
4453:
4364:
4299:
4235:
4185:
4116:
4056:
4021:
3966:
3924:
3897:
3862:
3836:
3816:
3733:
3649:
3456:
3423:
3390:
3353:
3259:
3201:
3143:
3091:
3021:
3010:
2871:
2767:
2692:
2648:
2619:
2575:
2532:
2499:
2419:
2283:
2246:
2195:
2163:
1893:
1748:
1721:
1685:
1611:
1578:
1530:
1470:
1443:
1417:
1168:
1145:
1125:
1103:
1068:
1042:
1007:
959:
821:
761:
730:
673:
634:
507:
422:
347:
317:
287:
258:
214:
185:
5292:Journal of Mathematical Analysis and Applications
3037:There shall correspond to each class or relation
1310:
1212:
5118:(Second ed.). John Wiley & Sons, Inc.
4300:{\displaystyle {\frac {dG(x)}{dx}}=-\delta (x)}
4193:and similarly the distributional derivative of
4128:of the Heaviside step function is equal to the
3650:{\displaystyle f_{\alpha }\in \mathbb {F} _{q}}
3035:
4186:{\displaystyle {\frac {dH(x)}{dx}}=\delta (x)}
3741:be their vanishing locus. Then, the function
1903:Expanding the product on the left hand side,
8:
5212:. Cambridge UK: Cambridge University Press.
5019:
5017:
5015:
5013:
5011:
4898:
4886:
2693:{\displaystyle \mathbf {1} _{A}(\omega )=0.}
2092:
2068:
1985:
1961:
1345:
1313:
1245:
1215:
954:
942:
502:
490:
4782:, which is the ultimate origin of the word
4236:{\displaystyle G(x):=\mathbf {1} _{x<0}}
4117:{\displaystyle H(x):=\mathbf {1} _{x>0}}
3484:Characteristic function in fuzzy set theory
3260:{\displaystyle \neg R(x_{1},\ldots x_{n}).}
3202:{\displaystyle \phi (x_{1},\ldots x_{n})=1}
3092:{\displaystyle \phi (x_{1},\ldots x_{n})=0}
2620:{\displaystyle \mathbf {1} _{A}(\omega )=1}
2429:This identity is used in a simple proof of
4765:
4763:
4606:
4562:
3526:, and the corresponding "sets" are called
5311:
5249:
5070:
4949:
4928:
4922:
4878:
4854:
4848:
4807:
4806:
4804:
4651:integrates to the numerical value of the
4649:inward normal derivative of the indicator
4623:
4611:
4598:
4591:
4585:
4573:
4567:
4549:
4548:
4543:
4536:
4523:
4518:
4516:
4508:
4494:
4490:
4489:
4487:
4478:
4438:
4437:
4432:
4425:
4412:
4407:
4392:
4383:
4377:
4365:{\displaystyle \delta _{S}(\mathbf {x} )}
4354:
4345:
4339:
4332:inward normal derivative of the indicator
4330:. Proceeding, it can be derived that the
4250:
4248:
4221:
4216:
4198:
4139:
4137:
4102:
4097:
4079:
4034:
4001:
3985:
3979:
3943:
3937:
3916:
3910:
3875:
3849:
3829:
3797:
3781:
3746:
3705:
3692:
3687:
3683:
3682:
3662:
3638:
3619:
3606:
3602:
3601:
3591:
3585:
3576:) continuous indicator function. Given a
3442:
3436:
3409:
3403:
3376:
3370:
3361:whenever any one of the functions equals
3339:
3320:
3307:
3301:
3245:
3229:
3214:
3184:
3168:
3156:
3132:
3116:
3104:
3074:
3058:
3046:
2933:
2928:
2909:
2904:
2892:
2806:
2801:
2789:
2729:
2724:
2712:
2669:
2664:
2661:
2632:
2596:
2591:
2588:
2569:
2568:
2553:
2548:
2545:
2521:
2520:
2512:
2481:
2480:
2468:
2384:
2370:
2355:
2350:
2343:
2327:
2322:
2310:
2275:
2270:
2267:
2239:
2188:
2180:
2178:
2153:
2143:
2138:
2133:
2119:
2111:
2110:
2055:
2040:
2030:
2025:
2020:
2012:
2004:
2003:
1954:
1933:
1923:
1918:
1913:
1910:
1880:
1870:
1865:
1860:
1842:
1832:
1821:
1816:
1801:
1796:
1791:
1769:
1763:
1740:
1734:
1708:
1663:
1658:
1653:
1631:
1625:
1595:
1570:
1551:
1545:
1519:
1514:
1496:
1491:
1486:
1483:
1462:
1456:
1436:
1402:
1397:
1387:
1382:
1372:
1367:
1357:
1352:
1338:
1333:
1323:
1318:
1316:
1291:
1286:
1272:
1267:
1257:
1252:
1239:
1234:
1224:
1219:
1193:
1188:
1183:
1181:
1158:
1138:
1118:
1089:
1084:
1081:
1055:
1028:
1023:
1020:
991:
940:
813:
807:
753:
748:
745:
724:
709:
704:
701:
654:
602:
567:
551:
533:
528:
525:
475:
470:
467:
384:
379:
376:
336:
330:
306:
300:
279:
274:
271:
235:
230:
227:
198:
162:
157:
154:
87:Learn how and when to remove this message
32:characteristic function (convex analysis)
50:This article includes a list of general
4989:
4759:
2211:. This is one form of the principle of
4795:The set of all indicator functions on
4022:{\displaystyle f_{\alpha }(x)^{q-1}=1}
3365:, it plays the role of logical OR: IF
731:{\displaystyle \mathbf {1} _{A}(x)\,.}
3144:{\displaystyle R(x_{1},\ldots x_{n})}
423:{\displaystyle \mathbf {1} _{A}(x)=.}
259:{\displaystyle \mathbf {1} _{A}(x)=0}
186:{\displaystyle \mathbf {1} _{A}(x)=1}
7:
3967:{\displaystyle f_{\alpha }(x)\neq 0}
2533:{\displaystyle A\in {\mathcal {F}},}
3514:(usually required to be at least a
2775:(also called "Fundamental Bridge").
1579:{\displaystyle A_{1},\dotsc ,A_{n}}
1104:{\displaystyle \mathbf {1} _{A}=0.}
1043:{\displaystyle \mathbf {1} _{A}=1.}
837:, which is defined as if using the
452:The indicator function of a subset
5180:. New York, NY: Raven Press Books.
4828:{\displaystyle {\mathcal {P}}(X),}
4689:Free variables and bound variables
4647:equal to one, it follows that the
4533:
4422:
3824:acts as an indicator function for
3216:
2993:
2978:
2954:
2851:
2827:
2750:
2714:
2562:
2490:
2474:
2399:
2393:
2374:
2312:
2247:{\displaystyle \operatorname {P} }
2241:
2056:
1427:and the indicator function of the
1069:{\displaystyle A\equiv \emptyset }
1063:
649:provides the equivalent notation,
56:it lacks sufficient corresponding
25:
436:is the indicator function of the
4574:
4550:
4544:
4519:
4509:
4439:
4433:
4408:
4393:
4355:
4217:
4098:
2929:
2905:
2802:
2725:
2665:
2592:
2549:
2351:
2323:
2284:{\displaystyle \mathbf {1} _{A}}
2271:
2134:
2021:
1914:
1861:
1817:
1792:
1729:that belong to none of the sets
1654:
1515:
1487:
1398:
1383:
1368:
1353:
1334:
1319:
1287:
1268:
1253:
1235:
1220:
1189:
1085:
1024:
762:{\displaystyle \mathbf {1} _{A}}
749:
705:
529:
471:
380:
365:of the property of belonging to
288:{\displaystyle \mathbf {1} _{A}}
275:
231:
158:
41:
5190:Introduction to Metamathematics
5029:Introduction to Metamathematics
2299:is equal to the probability of
5059:Journal of High Energy Physics
4960:
4819:
4813:
4603:
4595:
4513:
4505:
4397:
4389:
4359:
4351:
4294:
4288:
4265:
4259:
4209:
4203:
4180:
4174:
4154:
4148:
4090:
4084:
4045:
4039:
3998:
3991:
3955:
3949:
3886:
3880:
3794:
3787:
3757:
3751:
3717:
3711:
3644:
3612:
3251:
3222:
3190:
3161:
3138:
3109:
3080:
3051:
3005:
2999:
2990:
2984:
2972:
2960:
2948:
2945:
2939:
2921:
2915:
2900:
2866:
2863:
2857:
2842:
2839:
2833:
2821:
2818:
2812:
2797:
2762:
2756:
2744:
2741:
2735:
2720:
2681:
2675:
2608:
2602:
2565:
2540:the indicator random variable
2493:
2471:
2411:
2405:
2367:
2361:
2333:
2318:
2189:
2181:
2120:
2112:
2107:
2097:
2013:
2005:
2000:
1990:
1809:
1781:
1680:
1677:
1671:
1643:
1586:is a collection of subsets of
863:" has an unrelated meaning in
721:
715:
668:
656:
545:
539:
487:
414:
402:
396:
390:
247:
241:
174:
168:
1:
5251:10.1016/S0019-9958(65)90241-X
4944:for the set of all functions
3506:, or more generally, in some
3289:if the predicate is true and
3275:primitive recursive functions
2649:{\displaystyle \omega \in A,}
2456:Mean, variance and covariance
5352:Basic concepts in set theory
5304:10.1016/0022-247X(67)90189-8
1756:and is 0 otherwise. That is
4910:{\displaystyle Y=\{0,1\}=2}
4738:Dummy variable (statistics)
4684:Extension (predicate logic)
3925:{\displaystyle f_{\alpha }}
3457:{\displaystyle \phi _{n}=0}
3424:{\displaystyle \phi _{2}=0}
3391:{\displaystyle \phi _{1}=0}
3293:if the predicate is false.
2442:generalized Möbius function
829:is also used to denote the
357:The indicator function of
5378:
5151:Introduction to Algorithms
4778:
4743:Statistical classification
4674:Laplacian of the indicator
3548:Laplacian of the indicator
3545:
3538:like "tall", "warm", etc.
1050:By a similar argument, if
1008:{\displaystyle A\equiv X,}
865:classic probability theory
348:{\displaystyle \chi _{A}.}
29:
5244:(3). San Diego: 338–353.
4972:{\displaystyle f:X\to Y.}
4126:distributional derivative
4071:. For example, consider
2234:with probability measure
822:{\displaystyle \chi _{A}}
4873:This is a special case (
4643:By setting the function
4311:inward normal derivative
3534:seen in many real-world
3041:a representing function
1695:is clearly a product of
1540:More generally, suppose
896:probability distribution
892:characteristic functions
888:modern many-valued logic
869:traditional probabilists
798:Notation and terminology
145:is a subset of some set
5237:Information and Control
5210:Computability and Logic
5089:10.1007/JHEP11(2012)032
4799:can be identified with
4694:Heaviside step function
4073:Heaviside step function
3480:and the CASE function.
2436:In many cases, such as
1612:{\displaystyle x\in X:}
960:{\displaystyle \{0,1\}}
877:characteristic function
861:characteristic function
831:characteristic function
215:{\displaystyle x\in A,}
127:characteristic function
71:more precise citations.
18:Characteristic sequence
5114:Folland, G.B. (1999).
4973:
4938:
4911:
4867:
4866:{\displaystyle 2^{X}.}
4829:
4748:Zero-one loss function
4635:
4455:
4366:
4301:
4237:
4187:
4118:
4058:
4057:{\displaystyle P(x)=0}
4023:
3968:
3926:
3905:, otherwise, for some
3899:
3898:{\displaystyle P(x)=1}
3864:
3863:{\displaystyle x\in V}
3838:
3818:
3735:
3651:
3464:THEN their product is
3458:
3425:
3392:
3355:
3268:
3261:
3203:
3145:
3093:
3012:
2873:
2769:
2694:
2650:
2621:
2577:
2534:
2501:
2421:
2285:
2248:
2197:
2165:
1895:
1750:
1723:
1722:{\displaystyle x\in X}
1687:
1613:
1580:
1532:
1472:
1445:
1419:
1170:
1147:
1127:
1105:
1070:
1044:
1009:
961:
823:
763:
732:
696:to be used instead of
675:
636:
509:
424:
349:
319:
318:{\displaystyle I_{A},}
289:
260:
216:
187:
114:
5138:Leiserson, Charles E.
4974:
4939:
4937:{\displaystyle Y^{X}}
4912:
4868:
4830:
4636:
4456:
4367:
4302:
4238:
4188:
4119:
4059:
4024:
3974:, which implies that
3969:
3927:
3900:
3865:
3839:
3819:
3736:
3652:
3459:
3426:
3393:
3356:
3262:
3204:
3146:
3094:
3031:representing function
3013:
2874:
2770:
2695:
2651:
2622:
2578:
2535:
2502:
2422:
2286:
2249:
2198:
2166:
1896:
1751:
1749:{\displaystyle A_{k}}
1724:
1688:
1614:
1581:
1533:
1473:
1471:{\displaystyle A^{C}}
1446:
1420:
1171:
1148:
1128:
1106:
1071:
1045:
1010:
962:
890:, predicates are the
844:A related concept in
824:
769:is sometimes denoted
764:
733:
676:
637:
510:
425:
350:
320:
290:
261:
217:
188:
104:
5044:Course in Arithmetic
4948:
4921:
4877:
4847:
4803:
4477:
4376:
4338:
4247:
4197:
4136:
4130:Dirac delta function
4078:
4033:
3978:
3936:
3909:
3874:
3848:
3828:
3745:
3661:
3584:
3524:membership functions
3435:
3402:
3369:
3300:
3213:
3155:
3103:
3045:
2891:
2788:
2711:
2660:
2631:
2587:
2544:
2511:
2467:
2309:
2266:
2238:
2177:
1909:
1762:
1733:
1707:
1624:
1594:
1544:
1482:
1455:
1435:
1180:
1157:
1137:
1117:
1080:
1054:
1019:
990:
939:
806:
744:
700:
653:
524:
466:
375:
329:
299:
270:
226:
197:
153:
5206:Jeffrey, Richard C.
5081:2012JHEP...11..032L
4728:Membership function
4326:will be denoted by
3697:
3558:connected component
2431:Markov's inequality
2213:inclusion-exclusion
2196:{\displaystyle |F|}
1153:are two subsets of
867:. For this reason,
440:as a subset of the
5362:Types of functions
5357:Probability theory
5347:Mathematical logic
5313:10338.dmlcz/103980
4969:
4934:
4917:) of the notation
4907:
4863:
4825:
4631:
4451:
4362:
4297:
4233:
4183:
4114:
4054:
4019:
3964:
3922:
3895:
3860:
3834:
3814:
3731:
3681:
3647:
3562:algebraic geometry
3496:(non-members). In
3472:when the function
3454:
3421:
3388:
3351:
3257:
3199:
3141:
3089:
3008:
2869:
2765:
2690:
2646:
2617:
2573:
2530:
2497:
2496:
2417:
2281:
2244:
2224:probability theory
2193:
2161:
2148:
2096:
2035:
1989:
1928:
1891:
1875:
1837:
1780:
1746:
1719:
1683:
1642:
1609:
1576:
1528:
1468:
1441:
1415:
1413:
1169:{\displaystyle X,}
1166:
1143:
1123:
1101:
1066:
1040:
1005:
957:
873:indicator function
819:
759:
728:
671:
632:
627:
505:
434:Dirichlet function
420:
345:
315:
285:
256:
212:
183:
123:indicator function
115:
5337:Integral calculus
5219:978-0-521-00758-0
5165:978-0-262-03293-3
5142:Rivest, Ronald L.
5134:Cormen, Thomas H.
5125:978-0-471-31716-6
4718:Macaulay brackets
4713:identity relation
4699:Identity function
4322:. The surface of
4277:
4166:
3837:{\displaystyle V}
3568:, however, every
2462:probability space
2232:probability space
2139:
2051:
2026:
1950:
1919:
1866:
1828:
1765:
1627:
1444:{\displaystyle A}
1146:{\displaystyle B}
1126:{\displaystyle A}
621:
609:
605:
599:
586:
574:
570:
564:
432:For example, the
266:otherwise, where
97:
96:
89:
16:(Redirected from
5369:
5317:
5315:
5279:
5253:
5223:
5202:Burgess, John P.
5193:
5181:
5169:
5129:
5101:
5100:
5074:
5054:
5048:
5047:
5039:
5033:
5032:
5021:
5006:
5005:
4994:
4979:
4978:
4976:
4975:
4970:
4943:
4941:
4940:
4935:
4933:
4932:
4916:
4914:
4913:
4908:
4872:
4870:
4869:
4864:
4859:
4858:
4842:
4834:
4832:
4831:
4826:
4812:
4811:
4798:
4793:
4787:
4781:
4780:
4775:
4767:
4657:
4646:
4640:
4638:
4637:
4632:
4627:
4622:
4621:
4602:
4590:
4589:
4577:
4572:
4571:
4561:
4560:
4553:
4547:
4541:
4540:
4528:
4527:
4522:
4512:
4501:
4500:
4499:
4498:
4493:
4472:
4464:
4460:
4458:
4457:
4452:
4450:
4449:
4442:
4436:
4430:
4429:
4417:
4416:
4411:
4396:
4388:
4387:
4371:
4369:
4368:
4363:
4358:
4350:
4349:
4329:
4325:
4321:
4306:
4304:
4303:
4298:
4278:
4276:
4268:
4251:
4242:
4240:
4239:
4234:
4232:
4231:
4220:
4192:
4190:
4189:
4184:
4167:
4165:
4157:
4140:
4123:
4121:
4120:
4115:
4113:
4112:
4101:
4069:weak derivatives
4063:
4061:
4060:
4055:
4028:
4026:
4025:
4020:
4012:
4011:
3990:
3989:
3973:
3971:
3970:
3965:
3948:
3947:
3931:
3929:
3928:
3923:
3921:
3920:
3904:
3902:
3901:
3896:
3869:
3867:
3866:
3861:
3843:
3841:
3840:
3835:
3823:
3821:
3820:
3815:
3813:
3809:
3808:
3807:
3786:
3785:
3740:
3738:
3737:
3732:
3730:
3726:
3710:
3709:
3696:
3691:
3686:
3656:
3654:
3653:
3648:
3643:
3642:
3624:
3623:
3611:
3610:
3605:
3596:
3595:
3505:
3499:fuzzy set theory
3495:
3491:
3475:
3471:
3467:
3463:
3461:
3460:
3455:
3447:
3446:
3430:
3428:
3427:
3422:
3414:
3413:
3397:
3395:
3394:
3389:
3381:
3380:
3364:
3360:
3358:
3357:
3352:
3344:
3343:
3325:
3324:
3312:
3311:
3292:
3288:
3285:takes on values
3284:
3280:
3266:
3264:
3263:
3258:
3250:
3249:
3234:
3233:
3208:
3206:
3205:
3200:
3189:
3188:
3173:
3172:
3150:
3148:
3147:
3142:
3137:
3136:
3121:
3120:
3098:
3096:
3095:
3090:
3079:
3078:
3063:
3062:
3040:
3017:
3015:
3014:
3009:
2938:
2937:
2932:
2914:
2913:
2908:
2878:
2876:
2875:
2870:
2811:
2810:
2805:
2774:
2772:
2771:
2766:
2734:
2733:
2728:
2699:
2697:
2696:
2691:
2674:
2673:
2668:
2655:
2653:
2652:
2647:
2626:
2624:
2623:
2618:
2601:
2600:
2595:
2582:
2580:
2579:
2574:
2572:
2558:
2557:
2552:
2539:
2537:
2536:
2531:
2526:
2525:
2506:
2504:
2503:
2498:
2486:
2485:
2426:
2424:
2423:
2418:
2389:
2388:
2360:
2359:
2354:
2348:
2347:
2332:
2331:
2326:
2302:
2290:
2288:
2287:
2282:
2280:
2279:
2274:
2257:
2253:
2251:
2250:
2245:
2229:
2210:
2202:
2200:
2199:
2194:
2192:
2184:
2170:
2168:
2167:
2162:
2160:
2159:
2158:
2157:
2147:
2137:
2131:
2130:
2123:
2115:
2095:
2047:
2046:
2045:
2044:
2034:
2024:
2018:
2017:
2016:
2008:
1988:
1940:
1939:
1938:
1937:
1927:
1917:
1900:
1898:
1897:
1892:
1887:
1886:
1885:
1884:
1874:
1864:
1849:
1848:
1847:
1846:
1836:
1820:
1808:
1807:
1806:
1805:
1795:
1779:
1755:
1753:
1752:
1747:
1745:
1744:
1728:
1726:
1725:
1720:
1702:
1698:
1692:
1690:
1689:
1684:
1670:
1669:
1668:
1667:
1657:
1641:
1618:
1616:
1615:
1610:
1589:
1585:
1583:
1582:
1577:
1575:
1574:
1556:
1555:
1537:
1535:
1534:
1529:
1524:
1523:
1518:
1503:
1502:
1501:
1500:
1490:
1477:
1475:
1474:
1469:
1467:
1466:
1450:
1448:
1447:
1442:
1424:
1422:
1421:
1416:
1414:
1407:
1406:
1401:
1392:
1391:
1386:
1377:
1376:
1371:
1362:
1361:
1356:
1344:
1343:
1342:
1337:
1328:
1327:
1322:
1302:
1301:
1290:
1277:
1276:
1271:
1262:
1261:
1256:
1244:
1243:
1238:
1229:
1228:
1223:
1204:
1203:
1192:
1175:
1173:
1172:
1167:
1152:
1150:
1149:
1144:
1132:
1130:
1129:
1124:
1110:
1108:
1107:
1102:
1094:
1093:
1088:
1075:
1073:
1072:
1067:
1049:
1047:
1046:
1041:
1033:
1032:
1027:
1014:
1012:
1011:
1006:
985:
977:
970:This mapping is
966:
964:
963:
958:
931:
924:
920:
902:Basic properties
828:
826:
825:
820:
818:
817:
793:
789:
782:
775:
768:
766:
765:
760:
758:
757:
752:
737:
735:
734:
729:
714:
713:
708:
695:
693:
680:
678:
677:
674:{\displaystyle }
672:
642:
639:
638:
633:
631:
630:
619:
607:
606:
603:
597:
584:
572:
571:
568:
562:
538:
537:
532:
514:
512:
511:
506:
480:
479:
474:
459:
455:
438:rational numbers
429:
427:
426:
421:
389:
388:
383:
368:
360:
354:
352:
351:
346:
341:
340:
324:
322:
321:
316:
311:
310:
294:
292:
291:
286:
284:
283:
278:
265:
263:
262:
257:
240:
239:
234:
221:
219:
218:
213:
192:
190:
189:
184:
167:
166:
161:
148:
144:
112:
108:
92:
85:
81:
78:
72:
67:this article by
58:inline citations
45:
44:
37:
21:
5377:
5376:
5372:
5371:
5370:
5368:
5367:
5366:
5322:
5321:
5320:
5282:
5226:
5220:
5196:
5186:Kleene, Stephen
5184:
5178:The Undecidable
5172:
5166:
5146:Stein, Clifford
5132:
5126:
5113:
5109:
5104:
5056:
5055:
5051:
5041:
5040:
5036:
5025:Kleene, Stephen
5023:
5022:
5009:
5002:The Undecidable
4996:
4995:
4991:
4987:
4982:
4946:
4945:
4924:
4919:
4918:
4875:
4874:
4850:
4845:
4844:
4840:
4801:
4800:
4796:
4794:
4790:
4773:
4768:
4761:
4757:
4752:
4733:Simple function
4709:Kronecker delta
4704:Iverson bracket
4664:
4655:
4644:
4607:
4581:
4563:
4542:
4532:
4517:
4488:
4483:
4475:
4474:
4470:
4469:of the surface
4465:is the outward
4462:
4431:
4421:
4406:
4379:
4374:
4373:
4341:
4336:
4335:
4327:
4323:
4319:
4269:
4252:
4245:
4244:
4215:
4195:
4194:
4158:
4141:
4134:
4133:
4096:
4076:
4075:
4031:
4030:
3997:
3981:
3976:
3975:
3939:
3934:
3933:
3912:
3907:
3906:
3872:
3871:
3846:
3845:
3826:
3825:
3793:
3777:
3770:
3766:
3743:
3742:
3701:
3674:
3670:
3659:
3658:
3634:
3615:
3600:
3587:
3582:
3581:
3550:
3544:
3503:
3493:
3489:
3486:
3473:
3469:
3465:
3438:
3433:
3432:
3431:OR ... OR
3405:
3400:
3399:
3372:
3367:
3366:
3362:
3335:
3316:
3303:
3298:
3297:
3290:
3286:
3282:
3281:of a predicate
3278:
3241:
3225:
3211:
3210:
3180:
3164:
3153:
3152:
3128:
3112:
3101:
3100:
3070:
3054:
3043:
3042:
3038:
3024:
2927:
2903:
2889:
2888:
2800:
2786:
2785:
2723:
2709:
2708:
2663:
2658:
2657:
2629:
2628:
2590:
2585:
2584:
2547:
2542:
2541:
2509:
2508:
2465:
2464:
2458:
2450:Möbius function
2380:
2349:
2339:
2321:
2307:
2306:
2300:
2293:random variable
2269:
2264:
2263:
2255:
2236:
2235:
2227:
2208:
2175:
2174:
2149:
2132:
2106:
2036:
2019:
1999:
1929:
1912:
1907:
1906:
1876:
1859:
1838:
1815:
1797:
1790:
1760:
1759:
1736:
1731:
1730:
1705:
1704:
1700:
1696:
1659:
1652:
1622:
1621:
1592:
1591:
1587:
1566:
1547:
1542:
1541:
1513:
1492:
1485:
1480:
1479:
1458:
1453:
1452:
1433:
1432:
1412:
1411:
1396:
1381:
1366:
1351:
1332:
1317:
1303:
1285:
1282:
1281:
1266:
1251:
1233:
1218:
1205:
1187:
1178:
1177:
1155:
1154:
1135:
1134:
1115:
1114:
1083:
1078:
1077:
1052:
1051:
1022:
1017:
1016:
988:
987:
983:
978:is a non-empty
975:
937:
936:
929:
922:
918:
904:
835:convex analysis
809:
804:
803:
800:
791:
790:, or even just
788:
784:
781:
777:
774:
770:
747:
742:
741:
703:
698:
697:
683:
682:
651:
650:
647:Iverson bracket
626:
625:
600:
591:
590:
565:
552:
527:
522:
520:
469:
464:
463:
457:
453:
450:
378:
373:
372:
366:
363:Iverson bracket
358:
332:
327:
326:
302:
297:
296:
273:
268:
267:
229:
224:
223:
195:
194:
156:
151:
150:
146:
142:
110:
106:
93:
82:
76:
73:
63:Please help to
62:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
5375:
5373:
5365:
5364:
5359:
5354:
5349:
5344:
5339:
5334:
5332:Measure theory
5324:
5323:
5319:
5318:
5298:(1): 145–174.
5290:-fuzzy sets".
5284:Goguen, Joseph
5280:
5224:
5218:
5198:Boolos, George
5194:
5182:
5176:, ed. (1965).
5170:
5164:
5130:
5124:
5110:
5108:
5105:
5103:
5102:
5049:
5034:
5007:
5000:, ed. (1965).
4988:
4986:
4983:
4981:
4980:
4968:
4965:
4962:
4959:
4956:
4953:
4931:
4927:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4885:
4882:
4862:
4857:
4853:
4824:
4821:
4818:
4815:
4810:
4788:
4784:characteristic
4758:
4756:
4753:
4751:
4750:
4745:
4740:
4735:
4730:
4725:
4720:
4715:
4706:
4701:
4696:
4691:
4686:
4681:
4676:
4671:
4665:
4663:
4660:
4630:
4626:
4620:
4617:
4614:
4610:
4605:
4601:
4597:
4594:
4588:
4584:
4580:
4576:
4570:
4566:
4559:
4556:
4552:
4546:
4539:
4535:
4531:
4526:
4521:
4515:
4511:
4507:
4504:
4497:
4492:
4486:
4482:
4448:
4445:
4441:
4435:
4428:
4424:
4420:
4415:
4410:
4405:
4402:
4399:
4395:
4391:
4386:
4382:
4361:
4357:
4353:
4348:
4344:
4296:
4293:
4290:
4287:
4284:
4281:
4275:
4272:
4267:
4264:
4261:
4258:
4255:
4230:
4227:
4224:
4219:
4214:
4211:
4208:
4205:
4202:
4182:
4179:
4176:
4173:
4170:
4164:
4161:
4156:
4153:
4150:
4147:
4144:
4111:
4108:
4105:
4100:
4095:
4092:
4089:
4086:
4083:
4053:
4050:
4047:
4044:
4041:
4038:
4018:
4015:
4010:
4007:
4004:
4000:
3996:
3993:
3988:
3984:
3963:
3960:
3957:
3954:
3951:
3946:
3942:
3919:
3915:
3894:
3891:
3888:
3885:
3882:
3879:
3859:
3856:
3853:
3833:
3812:
3806:
3803:
3800:
3796:
3792:
3789:
3784:
3780:
3776:
3773:
3769:
3765:
3762:
3759:
3756:
3753:
3750:
3729:
3725:
3722:
3719:
3716:
3713:
3708:
3704:
3700:
3695:
3690:
3685:
3680:
3677:
3673:
3669:
3666:
3646:
3641:
3637:
3633:
3630:
3627:
3622:
3618:
3614:
3609:
3604:
3599:
3594:
3590:
3570:affine variety
3543:
3540:
3504:[0, 1]
3485:
3482:
3453:
3450:
3445:
3441:
3420:
3417:
3412:
3408:
3387:
3384:
3379:
3375:
3350:
3347:
3342:
3338:
3334:
3331:
3328:
3323:
3319:
3315:
3310:
3306:
3277:as a function
3256:
3253:
3248:
3244:
3240:
3237:
3232:
3228:
3224:
3221:
3218:
3198:
3195:
3192:
3187:
3183:
3179:
3176:
3171:
3167:
3163:
3160:
3140:
3135:
3131:
3127:
3124:
3119:
3115:
3111:
3108:
3088:
3085:
3082:
3077:
3073:
3069:
3066:
3061:
3057:
3053:
3050:
3029:described the
3023:
3020:
3019:
3018:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2936:
2931:
2926:
2923:
2920:
2917:
2912:
2907:
2902:
2899:
2896:
2886:
2880:
2879:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2809:
2804:
2799:
2796:
2793:
2783:
2777:
2776:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2732:
2727:
2722:
2719:
2716:
2706:
2689:
2686:
2683:
2680:
2677:
2672:
2667:
2645:
2642:
2639:
2636:
2616:
2613:
2610:
2607:
2604:
2599:
2594:
2583:is defined by
2571:
2567:
2564:
2561:
2556:
2551:
2529:
2524:
2519:
2516:
2495:
2492:
2489:
2484:
2479:
2476:
2473:
2457:
2454:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2387:
2383:
2379:
2376:
2373:
2369:
2366:
2363:
2358:
2353:
2346:
2342:
2338:
2335:
2330:
2325:
2320:
2317:
2314:
2297:expected value
2278:
2273:
2260:measurable set
2243:
2191:
2187:
2183:
2156:
2152:
2146:
2142:
2136:
2129:
2126:
2122:
2118:
2114:
2109:
2105:
2102:
2099:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2054:
2050:
2043:
2039:
2033:
2029:
2023:
2015:
2011:
2007:
2002:
1998:
1995:
1992:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1953:
1949:
1946:
1943:
1936:
1932:
1926:
1922:
1916:
1890:
1883:
1879:
1873:
1869:
1863:
1858:
1855:
1852:
1845:
1841:
1835:
1831:
1827:
1824:
1819:
1814:
1811:
1804:
1800:
1794:
1789:
1786:
1783:
1778:
1775:
1772:
1768:
1743:
1739:
1718:
1715:
1712:
1682:
1679:
1676:
1673:
1666:
1662:
1656:
1651:
1648:
1645:
1640:
1637:
1634:
1630:
1608:
1605:
1602:
1599:
1573:
1569:
1565:
1562:
1559:
1554:
1550:
1527:
1522:
1517:
1512:
1509:
1506:
1499:
1495:
1489:
1465:
1461:
1440:
1410:
1405:
1400:
1395:
1390:
1385:
1380:
1375:
1370:
1365:
1360:
1355:
1350:
1347:
1341:
1336:
1331:
1326:
1321:
1315:
1312:
1309:
1306:
1304:
1300:
1297:
1294:
1289:
1284:
1283:
1280:
1275:
1270:
1265:
1260:
1255:
1250:
1247:
1242:
1237:
1232:
1227:
1222:
1217:
1214:
1211:
1208:
1206:
1202:
1199:
1196:
1191:
1186:
1185:
1165:
1162:
1142:
1122:
1100:
1097:
1092:
1087:
1065:
1062:
1059:
1039:
1036:
1031:
1026:
1004:
1001:
998:
995:
956:
953:
950:
947:
944:
912:characteristic
903:
900:
854:bound variable
850:dummy variable
816:
812:
799:
796:
786:
779:
772:
756:
751:
727:
723:
720:
717:
712:
707:
670:
667:
664:
661:
658:
629:
624:
618:
615:
612:
604: if
601:
596:
593:
592:
589:
583:
580:
577:
569: if
566:
561:
558:
557:
555:
550:
547:
544:
541:
536:
531:
504:
501:
498:
495:
492:
489:
486:
483:
478:
473:
460:is a function
449:
446:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
387:
382:
344:
339:
335:
314:
309:
305:
282:
277:
255:
252:
249:
246:
243:
238:
233:
211:
208:
205:
202:
182:
179:
176:
173:
170:
165:
160:
95:
94:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5374:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5342:Real analysis
5340:
5338:
5335:
5333:
5330:
5329:
5327:
5314:
5309:
5305:
5301:
5297:
5293:
5289:
5285:
5281:
5277:
5273:
5269:
5265:
5261:
5257:
5252:
5247:
5243:
5239:
5238:
5233:
5230:(June 1965).
5229:
5225:
5221:
5215:
5211:
5207:
5203:
5199:
5195:
5191:
5187:
5183:
5179:
5175:
5174:Davis, Martin
5171:
5167:
5161:
5157:
5153:
5152:
5147:
5143:
5139:
5135:
5131:
5127:
5121:
5117:
5112:
5111:
5106:
5098:
5094:
5090:
5086:
5082:
5078:
5073:
5068:
5065:(11): 29–30.
5064:
5060:
5053:
5050:
5045:
5038:
5035:
5030:
5026:
5020:
5018:
5016:
5014:
5012:
5008:
5003:
4999:
4998:Davis, Martin
4993:
4990:
4984:
4966:
4963:
4957:
4954:
4951:
4929:
4925:
4904:
4901:
4895:
4892:
4889:
4883:
4880:
4860:
4855:
4851:
4838:
4822:
4816:
4792:
4789:
4785:
4772:
4766:
4764:
4760:
4754:
4749:
4746:
4744:
4741:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4719:
4716:
4714:
4710:
4707:
4705:
4702:
4700:
4697:
4695:
4692:
4690:
4687:
4685:
4682:
4680:
4677:
4675:
4672:
4670:
4669:Dirac measure
4667:
4666:
4661:
4659:
4654:
4650:
4641:
4628:
4624:
4618:
4615:
4612:
4608:
4599:
4592:
4586:
4582:
4578:
4568:
4564:
4557:
4554:
4537:
4529:
4524:
4502:
4495:
4484:
4480:
4468:
4446:
4443:
4426:
4418:
4413:
4403:
4400:
4384:
4380:
4346:
4342:
4333:
4316:
4312:
4307:
4291:
4285:
4282:
4279:
4273:
4270:
4262:
4256:
4253:
4228:
4225:
4222:
4212:
4206:
4200:
4177:
4171:
4168:
4162:
4159:
4151:
4145:
4142:
4131:
4127:
4109:
4106:
4103:
4093:
4087:
4081:
4074:
4070:
4065:
4051:
4048:
4042:
4036:
4016:
4013:
4008:
4005:
4002:
3994:
3986:
3982:
3961:
3958:
3952:
3944:
3940:
3917:
3913:
3892:
3889:
3883:
3877:
3857:
3854:
3851:
3831:
3810:
3804:
3801:
3798:
3790:
3782:
3778:
3774:
3771:
3767:
3763:
3760:
3754:
3748:
3727:
3723:
3720:
3714:
3706:
3702:
3698:
3693:
3688:
3678:
3675:
3671:
3667:
3664:
3639:
3635:
3631:
3628:
3625:
3620:
3616:
3607:
3597:
3592:
3588:
3580:of functions
3579:
3575:
3571:
3567:
3566:finite fields
3563:
3559:
3555:
3549:
3541:
3539:
3537:
3533:
3529:
3525:
3521:
3517:
3513:
3509:
3501:
3500:
3492:(members) or
3483:
3481:
3479:
3451:
3448:
3443:
3439:
3418:
3415:
3410:
3406:
3385:
3382:
3377:
3373:
3348:
3345:
3340:
3336:
3332:
3329:
3326:
3321:
3317:
3313:
3308:
3304:
3294:
3276:
3272:
3267:
3254:
3246:
3242:
3238:
3235:
3230:
3226:
3219:
3196:
3193:
3185:
3181:
3177:
3174:
3169:
3165:
3158:
3133:
3129:
3125:
3122:
3117:
3113:
3106:
3086:
3083:
3075:
3071:
3067:
3064:
3059:
3055:
3048:
3034:
3032:
3028:
3002:
2996:
2987:
2981:
2975:
2969:
2966:
2963:
2957:
2951:
2942:
2934:
2924:
2918:
2910:
2897:
2894:
2887:
2885:
2882:
2881:
2860:
2854:
2848:
2845:
2836:
2830:
2824:
2815:
2807:
2794:
2791:
2784:
2782:
2779:
2778:
2759:
2753:
2747:
2738:
2730:
2717:
2707:
2705:
2702:
2701:
2700:
2687:
2684:
2678:
2670:
2643:
2640:
2637:
2634:
2614:
2611:
2605:
2597:
2559:
2554:
2527:
2517:
2514:
2487:
2477:
2463:
2455:
2453:
2451:
2447:
2446:number theory
2443:
2439:
2434:
2432:
2427:
2414:
2408:
2402:
2396:
2390:
2385:
2381:
2377:
2371:
2364:
2356:
2344:
2340:
2336:
2328:
2315:
2304:
2298:
2294:
2276:
2261:
2233:
2225:
2221:
2220:combinatorics
2216:
2214:
2206:
2185:
2171:
2154:
2150:
2144:
2140:
2127:
2124:
2116:
2103:
2100:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2065:
2062:
2059:
2052:
2048:
2041:
2037:
2031:
2027:
2009:
1996:
1993:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1958:
1955:
1951:
1947:
1944:
1941:
1934:
1930:
1924:
1920:
1904:
1901:
1888:
1881:
1877:
1871:
1867:
1856:
1853:
1850:
1843:
1839:
1833:
1829:
1825:
1822:
1812:
1802:
1798:
1787:
1784:
1776:
1773:
1770:
1766:
1757:
1741:
1737:
1716:
1713:
1710:
1693:
1674:
1664:
1660:
1649:
1646:
1638:
1635:
1632:
1628:
1619:
1606:
1603:
1600:
1597:
1571:
1567:
1563:
1560:
1557:
1552:
1548:
1538:
1525:
1520:
1510:
1507:
1504:
1497:
1493:
1463:
1459:
1438:
1430:
1425:
1408:
1403:
1393:
1388:
1378:
1373:
1363:
1358:
1348:
1339:
1329:
1324:
1307:
1305:
1298:
1295:
1292:
1278:
1273:
1263:
1258:
1248:
1240:
1230:
1225:
1209:
1207:
1200:
1197:
1194:
1163:
1160:
1140:
1120:
1111:
1098:
1095:
1090:
1060:
1057:
1037:
1034:
1029:
1002:
999:
996:
993:
981:
980:proper subset
973:
968:
951:
948:
945:
935:
927:
916:
913:
909:
901:
899:
897:
893:
889:
885:
880:
878:
874:
871:use the term
870:
866:
862:
857:
855:
851:
848:is that of a
847:
842:
840:
836:
832:
814:
810:
802:The notation
797:
795:
754:
740:The function
738:
725:
718:
710:
691:
687:
665:
662:
659:
648:
643:
641:
622:
616:
613:
610:
594:
587:
581:
578:
575:
559:
553:
548:
542:
534:
518:
515:
499:
496:
493:
484:
481:
476:
461:
447:
445:
443:
439:
435:
430:
417:
411:
408:
405:
399:
393:
385:
370:
364:
355:
342:
337:
333:
312:
307:
303:
280:
253:
250:
244:
236:
209:
206:
203:
200:
180:
177:
171:
163:
140:
136:
132:
128:
124:
120:
103:
99:
91:
88:
80:
77:December 2009
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
5295:
5291:
5287:
5241:
5235:
5232:"Fuzzy sets"
5209:
5189:
5177:
5150:
5115:
5062:
5058:
5052:
5046:. p. 5.
5043:
5037:
5028:
5001:
4992:
4791:
4783:
4771:Greek letter
4653:surface area
4642:
4314:
4310:
4308:
4066:
3551:
3527:
3497:
3487:
3478:mu operators
3295:
3269:
3036:
3030:
3025:
2459:
2438:order theory
2435:
2428:
2305:
2217:
2172:
1905:
1902:
1758:
1694:
1620:
1539:
1426:
1112:
969:
928:elements of
921:of some set
917:of a subset
911:
907:
905:
881:
876:
872:
858:
843:
801:
739:
689:
685:
644:
519:
516:
462:
451:
442:real numbers
431:
371:
369:; that is,
356:
126:
122:
116:
98:
83:
74:
55:
5228:Zadeh, L.A.
4679:Dirac delta
2205:cardinality
884:fuzzy logic
517:defined as
119:mathematics
69:introducing
5326:Categories
5268:0139.24606
4985:References
3932:, we have
3578:finite set
3572:admits a (
3560:. In the
3546:See also:
3542:Smoothness
3536:predicates
3027:Kurt Gödel
2884:Covariance
2656:otherwise
2291:becomes a
1590:. For any
1429:complement
974:only when
972:surjective
859:The term "
846:statistics
839:reciprocal
448:Definition
52:references
5286:(1967). "
5276:Q25938993
5260:0019-9958
5188:(1971) .
5072:1302.0864
5027:(1971) .
4961:→
4837:power set
4625:β
4616:−
4600:β
4583:∮
4555:∈
4534:∇
4530:⋅
4485:∫
4481:−
4444:∈
4423:∇
4419:⋅
4404:−
4381:δ
4343:δ
4286:δ
4283:−
4172:δ
4006:−
3987:α
3959:≠
3945:α
3918:α
3855:∈
3802:−
3783:α
3775:−
3764:∏
3707:α
3679:∈
3629:…
3598:∈
3593:α
3512:structure
3440:ϕ
3407:ϕ
3374:ϕ
3337:ϕ
3333:∗
3330:⋯
3327:∗
3318:ϕ
3314:∗
3305:ϕ
3239:…
3217:¬
3178:…
3159:ϕ
3126:…
3068:…
3049:ϕ
2997:
2982:
2976:−
2967:∩
2958:
2943:ω
2919:ω
2898:
2855:
2849:−
2831:
2816:ω
2795:
2754:
2739:ω
2718:
2679:ω
2638:∈
2635:ω
2606:ω
2566:→
2563:Ω
2560::
2518:∈
2475:Ω
2403:
2382:∫
2341:∫
2316:
2141:⋂
2101:−
2084:…
2066:⊆
2060:≠
2057:∅
2053:∑
2028:⋂
1994:−
1977:…
1959:⊆
1952:∑
1948:−
1921:⋃
1868:⋃
1857:−
1830:⋃
1826:−
1788:−
1774:∈
1767:∏
1714:∈
1650:−
1636:∈
1629:∏
1601:∈
1561:…
1511:−
1498:∁
1394:⋅
1379:−
1296:∪
1264:⋅
1198:∩
1064:∅
1061:≡
997:≡
908:indicator
811:χ
663:∈
614:∉
579:∈
488:→
482::
456:of a set
409:∈
334:χ
204:∈
5272:Wikidata
5208:(2002).
5097:56188533
4779:χαρακτήρ
4723:Multiset
4662:See also
4315:boundary
4029:, hence
2781:Variance
2460:Given a
915:function
139:function
5107:Sources
5077:Bibcode
5042:Serre.
4313:at the
4132:, i.e.
3574:Zariski
3554:support
3520:lattice
3508:algebra
2262:, then
2203:is the
932:to the
361:is the
149:, then
65:improve
5274:
5266:
5258:
5216:
5162:
5122:
5095:
4774:χ
4467:normal
4461:where
3532:degree
3271:Kleene
2448:, the
2295:whose
2173:where
1699:s and
1076:then
778:χ
620:
608:
598:
585:
573:
563:
131:subset
54:, but
5158:–99.
5093:S2CID
5067:arXiv
4755:Notes
3870:then
3844:. If
3556:is a
3528:fuzzy
3516:poset
2507:with
2258:is a
2230:is a
2226:: if
1451:i.e.
1176:then
1015:then
986:. If
934:range
894:of a
137:is a
133:of a
129:of a
125:or a
121:, an
5256:ISSN
5214:ISBN
5160:ISBN
5120:ISBN
5063:2012
4835:the
4769:The
4226:<
4124:The
4107:>
4064:.
3657:let
3151:and
2704:Mean
2254:and
1478:is:
1133:and
926:maps
906:The
886:and
645:The
325:and
222:and
5308:hdl
5300:doi
5264:Zbl
5246:doi
5085:doi
4839:of
4243:is
3564:of
3518:or
3510:or
3398:OR
3209:if
3099:if
2895:Cov
2792:Var
2627:if
2207:of
1431:of
1311:max
1213:min
1113:If
982:of
910:or
882:In
856:.)
833:in
681:or
193:if
135:set
117:In
5328::
5306:.
5296:18
5294:.
5270:.
5262:.
5254:.
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5204:;
5200:;
5156:94
5144:;
5140:;
5136:;
5091:.
5083:.
5075:.
5061:.
5010:^
4762:^
4658:.
4372::
4213::=
4094::=
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2433:.
2303::
2215:.
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1038:1.
967:.
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783:,
776:,
688:∈
549::=
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113:).
5316:.
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5302::
5288:L
5278:.
5248::
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5222:.
5168:.
5128:.
5099:.
5087::
5079::
5069::
4967:.
4964:Y
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4902:=
4899:}
4896:1
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4890:0
4887:{
4884:=
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4861:.
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4786:.
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4629:.
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3603:F
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3341:n
3322:2
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3255:.
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3194:=
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3186:n
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3052:(
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3003:B
3000:(
2994:P
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2757:(
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2685:=
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2412:)
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2014:|
2010:F
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2001:)
1997:1
1991:(
1986:}
1983:n
1980:,
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1971:2
1968:,
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1935:k
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1777:I
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1717:X
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1404:B
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1359:A
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1349:=
1346:}
1340:B
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1325:A
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1314:{
1308:=
1299:B
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1241:B
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1195:A
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1096:=
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1003:,
1000:X
994:A
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955:}
952:1
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946:0
943:{
930:X
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785:K
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750:1
726:.
722:)
719:x
716:(
711:A
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692:⟧
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686:x
684:⟦
669:]
666:A
660:x
657:[
623:.
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